Embedded newform 560.2.cp.a.221.61

• Label: 560.2.cp.a.221.61
• Level: $560$
• Weight: $2$
• Character: 560.221
• Analytic conductor: $4.472$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	560.cp (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.47162251319\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(64\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			221.61
Character	\(\chi\)	\(=\)	560.221
Dual form			560.2.cp.a.261.61

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.37298 - 0.338996i) q^{2} +(-2.33375 + 0.625325i) q^{3} +(1.77016 - 0.930872i) q^{4} +(-0.965926 - 0.258819i) q^{5} +(-2.99221 + 1.64969i) q^{6} +(-2.28738 + 1.32962i) q^{7} +(2.11484 - 1.87815i) q^{8} +(2.45726 - 1.41870i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 4 q^{4}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\)	\(241\)	\(337\)	\(351\)	\(421\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(1\)	\(e\left(\frac{3}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.37298	−	0.338996i	0.970845	−	0.239707i
							\(3\)	−2.33375	+	0.625325i	−1.34739	+	0.361032i	−0.859170	−	0.511690i	\(-0.829020\pi\)
−0.488219	+	0.872721i	\(0.662353\pi\)
\(5\)	−0.965926	−	0.258819i	−0.431975	−	0.115747i
							\(7\)	−2.28738	+	1.32962i	−0.864548	+	0.502550i
\(11\)	0.172513	+	0.643828i	0.0520147	+	0.194121i								0.987044	−	0.160448i	\(-0.0512940\pi\)
							−0.935030	+	0.354570i	\(0.884627\pi\)
\(13\)	4.46689	+	4.46689i	1.23889	+	1.23889i	0.960455	+	0.278436i	\(0.0898162\pi\)
							0.278436	+	0.960455i	\(0.410184\pi\)
\(17\)	−3.07576	+	5.32737i	−0.745981	+	1.29208i	0.203754	+	0.979022i	\(0.434686\pi\)
							−0.949735	+	0.313055i	\(0.898648\pi\)
\(19\)	−1.30140	+	4.85689i	−0.298561	+	1.11425i	0.639786	+	0.768553i	\(0.279023\pi\)
							−0.938348	+	0.345693i	\(0.887644\pi\)
\(23\)	−6.06786	+	3.50328i	−1.26524	+	0.730485i	−0.974083	−	0.226192i	\(-0.927372\pi\)
							−0.291154	+	0.956676i	\(0.594039\pi\)
\(29\)	−0.261977	−	0.261977i	−0.0486479	−	0.0486479i	0.682364	−	0.731012i	\(-0.260952\pi\)
							−0.731012	+	0.682364i	\(0.760952\pi\)
\(31\)	−1.21140	+	2.09821i	−0.217574	+	0.376850i	−0.954066	−	0.299597i	\(-0.903148\pi\)
							0.736492	+	0.676447i	\(0.236481\pi\)
\(37\)	2.70341	+	0.724376i	0.444438	+	0.119087i	0.474096	−	0.880473i	\(-0.342775\pi\)
							−0.0296579	+	0.999560i	\(0.509442\pi\)
\(41\)		−	5.15441i		−	0.804984i	−0.915424	−	0.402492i	\(-0.868144\pi\)
							0.915424	−	0.402492i	\(-0.131856\pi\)
\(43\)	−1.41116	+	1.41116i	−0.215200	+	0.215200i	−0.806472	−	0.591272i	\(-0.798626\pi\)
							0.591272	+	0.806472i	\(0.298626\pi\)
\(47\)	−3.07074	−	5.31867i	−0.447913	−	0.775808i	0.550337	−	0.834942i	\(-0.314499\pi\)
							−0.998250	+	0.0591348i	\(0.981166\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.cp.a.221.61	✓	256
7.2	even	3	inner	560.2.cp.a.541.17	yes	256
16.5	even	4	inner	560.2.cp.a.501.17	yes	256
112.37	even	12	inner	560.2.cp.a.261.61	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.cp.a.221.61	✓	256	1.1	even	1	trivial
560.2.cp.a.261.61	yes	256	112.37	even	12	inner
560.2.cp.a.501.17	yes	256	16.5	even	4	inner
560.2.cp.a.541.17	yes	256	7.2	even	3	inner